Simplifying trigonometric equationsHere is the question: simplify the expression sin ⁡ ( f + g ) + sin ⁡ ( f − g ) cos ⁡ ( f + g ) + cos ⁡ ( f − g ) .For this questions, are all of the addition and subtraction identities of sin and cos required? I am not certain how to approach this question. If someone could help it would be very appreciated!

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Simplifying trigonometric equationsHere is the question: simplify the expression            sin      ⁡      (      f      +      g      )      +      sin      ⁡      (      f      −      g      )              cos      ⁡      (      f      +      g      )      +      cos      ⁡      (      f      −      g      )        .For this questions, are all of the addition and subtraction identities of sin and cos required? I am not certain how to approach this question. If someone could help it would be very appreciated!

elulamami · Accepted Answer

Okay  sin  ⁡  (  f  +  g  )  =  sin  ⁡  f  cos  ⁡  g  +  sin  ⁡  g  cos  ⁡  f  sin  ⁡  (  f  −  g  )  =  sin  ⁡  f  cos  ⁡  g  −  sin  ⁡  g  cos  ⁡  f  cos  ⁡  (  f  +  g  )  =  cos  ⁡  f  cos  ⁡  g  −  sin  ⁡  f  sin  ⁡  g  cos  ⁡  (  f  −  g  )  =  cos  ⁡  f  cos  ⁡  g  +  sin  ⁡  f  sin  ⁡  g  sin  ⁡  (  f  +  g  )  +  sin  ⁡  (  f  −  g  )  =  sin  ⁡  f  cos  ⁡  g  +  sin  ⁡  g  cos  ⁡  f  +  sin  ⁡  f  cos  ⁡  g  −  sin  ⁡  g  cos  ⁡  f  =  2  sin  ⁡  f  cos  ⁡  g  cos  ⁡  (  f  +  g  )  +  cos  ⁡  (  f  −  g  )  =  cos  ⁡  f  cos  ⁡  g  −  sin  ⁡  g  sin  ⁡  f  +  cos  ⁡  f  cos  ⁡  g  +  sin  ⁡  f  sin  ⁡  g  =  2  cos  ⁡  f  cos  ⁡  gSo dividing gives            2      sin      ⁡      f      cos      ⁡      g              2      cos      ⁡      f      cos      ⁡      g        =            sin      ⁡      f      cos      ⁡      g              cos      ⁡      f      cos      ⁡      g        =            sin      ⁡      f              cos      ⁡      f        =  tan  ⁡  f

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